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Theorem reupick3 4274
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reupick3 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reupick3
StepHypRef Expression
1 df-reu 3140 . . . 4 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 df-rex 3139 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
3 anass 472 . . . . . 6 (((𝑥𝐴𝜑) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
43exbii 1849 . . . . 5 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
52, 4bitr4i 281 . . . 4 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥((𝑥𝐴𝜑) ∧ 𝜓))
6 eupick 2721 . . . 4 ((∃!𝑥(𝑥𝐴𝜑) ∧ ∃𝑥((𝑥𝐴𝜑) ∧ 𝜓)) → ((𝑥𝐴𝜑) → 𝜓))
71, 5, 6syl2anb 600 . . 3 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓)) → ((𝑥𝐴𝜑) → 𝜓))
87expd 419 . 2 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓)) → (𝑥𝐴 → (𝜑𝜓)))
983impia 1114 1 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wex 1781  wcel 2115  ∃!weu 2654  wrex 3134  ∃!wreu 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-rex 3139  df-reu 3140
This theorem is referenced by:  reupick2  4275  fvineqsneq  34777
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